No, I do not believe that your engine as designed would possibly operate for very long without coolant in the water jackets. Even if ordinary engine oil was poured in there in place of the water, it would not work very well as others have explained.
I don't disagree with any of the answers listed to this point. However, I would say that once one begins to design such an oil cooled engine, they would soon find that it needed to have a separate system in order to be able to protect the engine from the contaminants introduced into oil due to cooling because that oil suffers greatly at the higher temperatures. Then they would discover that the oil used for cooling needs to have better heat transfer to work efficiently and to keep oil from getting scorched, so that with the separate systems they could make it more "water-like" in characteristics. Keep going with this (based almost completely on the other answers) and they will end up with an "oil" that is much like a 50-50 anti-freeze - which is what we currently use. This mixture uses other chemicals to enhance its abilities to withstand cold temperatures and reducing effects of corrosion while keeping most of the properties of the water intact and that is what the oil would have to do to overcome problems it faces.
So why do air-cooled engines exist? Probably because the weight of the material needed to provide the coolant jacket was greater than the cost of the materials for the 'fins' and had no significant advantage in cooling as long as there was oil circulating in sufficient enough quantities to help out. There is little doubt that air cooling has a difficult time to maintain uniform temperatures compared to liquid coolants, but the weight advantage of air using fins versus coolant using a coolant jacket is undeniable. It is made less if the coolant cooled engine uses the same types of materials as the air cooled engine, but that makes a large engine much more expensive. Somebody made the trade-off a long time ago and it has proven out over the years.
Who knows, keep thinking about it - someone might come up with a better cooling method for engines like these. Just look what they have done with electric motors over the years. Huge motors are now quite small with better torque response, etc. The same has happened to internal combustion engines though not quite as dramatically so far.